Generation of stochastic electromagnetic beams with complete controllable coherence

Xudong Chen; Chengcheng Chang; Ziyang Chen; Zhili Lin; Jixiong Pu

doi:10.1364/OE.24.021587

1. Introduction

The modulation of the spatial coherence of light has long been a subject of great importance in a broad area of optics. It is well known that, contrary to spatially coherent monochromatic light, partially coherent light (PCL) is able to suppress speckle noise and improve image quality [1]. PCL has also been suggested to alleviate the harmful effects in optical propagation through random or turbulent media [2] and for optical communications [3]. Additionally, light sources with controlled spatial coherence could be useful in applications which require special spatial correlations, such as in generating propagation-invariant fields [4] and in advanced optical coherence tomography [5].

Until now there have been many papers concerning with the generation and propagation of PCL. In the scalar domain, many coherent models have been well studied such as Gaussian and multi-Gaussian Schell-model sources [6, 7], Bessel-Gaussian and Laguerre-Gaussian Schell-model sources [8], cosine-Gaussian Schell-models sources [9] and non-uniformly correlated sources [10, 11]. Based on the theory for devising genuine cross-spectral density (CSD) matrices for a stochastic electromagnetic beam (SEB) [12, 13], several scalar models have been also extended to the electromagnetic domain, such as the electromagnetic Multi-Gaussian Schell-model sources [14], electromagnetic non-uniformly correlated sources [15] and electromagnetic cosine-Gaussian Schell-model sources [16].

The SEBs can be generated with the help of two liquid-crystal (LC) spatial light modulators (SLMs) in a Mach-Zehnder interferometer [17,18]. However, to the best of our knowledge, until now, there is no investigation on complete control over the spatial coherence of a SEB, that is, the independent control of coherence degree along two mutually perpendicular directions.

In the present paper, we theoretically and experimentally investigate the generation of a SEB with complete controllable coherence. The coherence characteristic of the SEB are controlled by varying the magnitude of the phase modulation of two crossed phase only SLMs (PSLMs). We measure the coherence properties and the beam propagation factor M² of the beam. The experimental results confirm well our theoretical predictions.

2. Theoretical analysis

Let us consider a planar, statistically stationary electric field in the source plane. We may characterize the second-order correlation properties of the source by the CSD matrix [19]

\overset{\leftrightarrow}{W} (r_{1}, r_{2}, ω) = (\begin{matrix} W_{x x} (r_{1}, r_{2}, ω) & W_{x y} (r_{1}, r_{2}, ω) \\ W_{y x} (r_{1}, r_{2}, ω) & W_{y y} (r_{1}, r_{2}, ω) \end{matrix}),

where,

r_{1}

and

r_{2}

are the two-dimensional position vector of two points Q1 and Q2 in the source plane,

ω

denotes the angular frequency and

W_{j k} (r_{1}, r_{2}, ω) = 〈 E_{j}^{*} (r_{1}, ω) E_{k} (r_{2}, ω) 〉 (j = x, y; k = x, y),

where E_j and E_k denote the components of the electric field in two mutually orthogonal x- and y-directions, perpendicular to the beam axis (z-axis); the asterisk denotes the complex conjugate and < > indicates an ensemble average in the space-frequency domain [20].

In what follows, we will omit the angular frequency dependence of all the quantities of interest for brevity. As we know, the coherence degree between the two components can be expressed as

μ_{j k} (r_{1}, r_{2}) = \frac{W_{j k} (r_{1}, r_{2})}{\sqrt{W_{j j} (r_{1})} \sqrt{W_{k k} (r_{2})}} .

The electric field component of SEB may be described by

E_{j} (r) = A_{j} (r) \exp (i ϕ_{j})

, where A is the real amplitude and ϕ is the phase. It has been reported that there is little variance in the reflectivity of PSLMs as a function of phase, that is, the amplitude effect during phase modulation is negligible [21]. Therefore, it is reasonable to assume that the real amplitude A is deterministic but the phase ϕ is random modulated on the PSLM plane. In order to simplify the analysis, we assume that

A_{x} (r_{1}) = A_{y} (r_{1}) \equiv A_{1}; A_{x} (r_{2}) = A_{y} (r_{2}) \equiv A_{2}

. Therefore, on substituting Eq. (2) into Eq. (3), we obtain the correlation of the field at these two points [22] expressed as

μ_{j k} (r_{1}, r_{2}) = 〈 \exp {- i [ϕ_{j} (r_{1}) - ϕ_{k} (r_{2})]} 〉 .

Taking into account Eqs. (3) and (4), the cross-spectral density matrix can be changed to

\begin{array}{l} \overset{\leftrightarrow}{W} (r_{1}, r_{2}) = A_{1} A_{2} (\begin{matrix} μ_{x x} (r_{1}, r_{2}) & μ_{x y} (r_{1}, r_{2}) \\ μ_{y x} (r_{1}, r_{2}) & μ_{y y} (r_{1}, r_{2}) \end{matrix}) \\ = A_{1} A_{2} (\begin{matrix} 〈 \exp {- i [ϕ_{x} (r_{1}) - ϕ_{x} (r_{2})]} 〉 & 〈 \exp {- i [ϕ_{x} (r_{1}) - ϕ_{y} (r_{2})]} 〉 \\ 〈 \exp {- i [ϕ_{y} (r_{1}) - ϕ_{x} (r_{2})]} 〉 & 〈 \exp {- i [ϕ_{y} (r_{1}) - ϕ_{y} (r_{2})]} 〉 \end{matrix}) . \end{array}

As we know, PSLM has the property that only the component of the incident electric field parallel to the LC director is phase modulated, whereas the component perpendicular to the LC director cannot be modulated [23]. In order to generate a SEB, we shall impose phase fluctuation upon both the orthogonally polarized part of the laser beam. Therefore, we utilize two crossed PSLMs, whose LC directors are parallel to the polarization of the incident complete coherent fields, to modulate the two orthogonal polarization components respectively. Suppose that $ϕ_{x} (r)$ and $ϕ_{y} (r)$ represent the random phase imposed on the electric field along x- and y- polarization and both of them follow the uniform probability distribution density

p [ϕ_{j} (r)] = {\begin{matrix} \frac{1}{2 π γ_{j} (r)} & for π - π γ_{j} (r) < ϕ_{j} (r) < π + π γ_{j} (r) \\ 0 & otherwise \end{matrix}, (j = x, y)

where

γ_{j} (r)

is referred to as modulation magnitude, which is located at

0 \leq γ_{j} (r) \leq 1

. It should be noted that the phase of every point equals to π when

γ_{j} (r) = 0

. Hence, a different value of

γ_{j} (r)

represents a different range of the random phase imposed on the point

r

. By making use of Eq. (6), we can find

\begin{array}{l} 〈 \exp [- i ϕ_{j} (r)] 〉 = \int_{- \infty}^{\infty} \exp [- i ϕ_{j} (r)] \cdot p [ϕ_{j} (r)] d ϕ_{j} \\ = \frac{1}{2 π γ_{j} (r)} \int_{π - π γ_{j} (r)}^{π + π γ_{j} (r)} \exp [- i ϕ_{j} (r)] d ϕ_{j}, \\ = - \sin c [γ_{j} (r)] \end{array}

and analogously

〈 \exp [i ϕ_{j} (r)] 〉 = - \sin c [γ_{j} (r)],

where

sinc(t)

is the normalized sinc function and equals 1 for t = 0 and

sin(πt)/πt

otherwise. Since the phase modulation for different points

r_{1}

and

r_{2}

are statistically independent, taking into account Eq. (7), one can obtain

\begin{array}{l} 〈 \exp [- i ϕ_{j} (r_{1})] \exp [i ϕ_{k} (r_{2})] 〉 \\ = {\begin{matrix} 1 & for r_{1} = r_{2} and j = k \\ 〈 exp [-i ϕ_{j} (r_{1})] 〉 〈 exp [i ϕ_{k} (r_{2})] 〉 & otherwise \end{matrix} . \end{array}

Then substituting Eq. (8) into Eq. (4), we obtain the mutual correlation distribution of the beam, which can be expressed as follows

\begin{array}{l} μ_{j k} (r_{1}, r_{2}) \\ = {\begin{matrix} 1 & for r_{1} = r_{2} and j = k (j = x, y; k = x, y) \\ \sin c [γ_{j} (r_{1})] \sin c[γ_{k} (r_{2})] & otherwise \end{matrix} . \end{array}

Equation (9) is the expression for measuring four correlations between two positions $r_{1}$ and $r_{2}$ . According to Eq. (9), we can use SLMs to control the coherence degree of the SEB by varying the modulation magnitude $γ (r)$ of the two points. If both $γ_{j} (r_{1})$ and $γ_{k} (r_{2})$ are equal to zero, $μ_{j k} (r_{1}, r_{2})$ will be equal to 1, that is, the fields at the two points are coherent; On the other hand, if either $γ_{j} (r_{1})$ or $γ_{k} (r_{2})$ are equal to 1, $μ_{j k} (r_{1}, r_{2})$ will be equal to 0 implying that the fields are incoherent [22].

When the modulation magnitude $γ$ is a function of the position $r$ , the modulation is spatial dependent, that is, the correlation structure of the SEB generated is non-uniform. On the other hand, if $γ$ is a constant, which is independent on the position, the range of the random phase is uniform in the entire light spot, that is, the modulation generates an evenly distributed spatial phase-correlation. Because of the limitation of length, we will only discuss the second case here. If $γ_{j} (r) \equiv γ_{j}, γ_{k} (r) \equiv γ_{k}$ , Eq. (9) can be simplified to be

μ_{j k} (r_{1}, r_{2}) = {\begin{matrix} 1 & for r_{1} = r_{2} and j = k \\ \sin c [γ_{j}] \sin c [γ_{k}] & otherwise \end{matrix} .

According to Eq. (10), the coherence degree of any two points is constant regardless of the distance between them. Substituting Eq. (10) into Eq. (5), we get

\overset{\leftrightarrow}{W} (r_{1}, r_{2}) = A_{1} A_{2} \cdot (\begin{matrix} {[\sin c (γ_{x})]}^{2} & \sin c (γ_{x}) sinc (γ_{y}) \\ \sin c (γ_{y}) \sin c (γ_{x}) & {[\sin c (γ_{y})]}^{2} \end{matrix}),

and

\overset{\leftrightarrow}{W} (r, r) = A_{1} A_{2} \cdot (\begin{matrix} 1 & \sin c (γ_{x}) \sin c (γ_{y}) \\ \sin c (γ_{y}) \sin c (γ_{x}) & 1 \end{matrix})

As can be seen from Eqs. (10) and (11), we are able to control the coherence degree of x - polarized component and y - polarized component, independently on each other, by adjusting the modulation magnitude $γ_{x}$ and $γ_{y}$ . Moreover, the spatial correlation structure of the SEB is uniform, that is, the coherence degree of the SEB does not depend neither on positions nor the distance between any two positions.

According to references [19] and [24], the degree of coherence of the electromagnetic field is defined by the formula

η (r_{1}, r_{2}) = \frac{Tr \overset{\leftrightarrow}{W} (r_{1}, r_{2})}{{[Tr \overset{\leftrightarrow}{W} (r_{1}, r_{1})Tr \overset{\leftrightarrow}{W} (r_{2}, r_{2})]}^{1/2}} = \frac{{[\sin {c(β}_{x})]}^{2} +[\sin {c(β}_{y} {)]}^{2}}{2} .

We can also obtain the degree of polarization of the electromagnetic field according to Wolf [19], expressed as follows

P (r) = {(1- \frac{4det \overset{\leftrightarrow}{W} (r, r)}{{[Tr \overset{\leftrightarrow}{W} (r, r)]}^{2}})}^{1/2} = \sin {c(β}_{x}) \sin {c(β}_{y}) .

There are several different approaches to the electromagnetic degree of coherence. One is the “degree of coherence for electromagnetic fields” [25], defined by

μ_{12} = {[\frac{Tr (\overset{\leftrightarrow}{W} (r_{1}, r_{2}) {\overset{\leftrightarrow}{W}}^{†} (r_{1}, r_{2}))}{I_{1} I_{2}}]}^{1 / 2} .

where

I_{1} =Tr \overset{\leftrightarrow}{W} (r_{1}, r_{1})

and

I_{2} =Tr \overset{\leftrightarrow}{W} (r_{2}, r_{2})

are the intensities at positions 1 and 2, respectively, and the dagger denotes the Hermitian adjoint. Substituting Eqs. (11) and (12) into Eq. (15), we get that the value of “degree of coherence for electromagnetic fields” is the same as Eq. (13). It was reported in [22] that the “intrinsic degrees of coherence” introduced in [26] equal unity or zero in the case of full phase correlation or full phase uncorrelation, respectively. This same result applies to the SEB we have generated, since we obtain the correlation of fields according to [22].

3. Experimental setup and results

We employ an optical arrangement for producing a SEB, as shown in Fig. 1. We use a Melles Griot He-Ne laser beam, linear polarized along the x direction, as a coherent light source. We expand the laser beam to 4mm in diameter, and then rotate the direction of polarization by 45° with a half-wave-plate (HWP). We use a non-polarizing beam splitter (NPBS) to split the laser beam into two beams with equal intensity. The beams become orthogonally polarized, after passing through two crossed polarizers, and then incident on two crossed SLMs, respectively. As a result, SLM1 is illuminated by the y-polarized light while SLM2 is illuminated by the x-polarized light. The SLMs used are two phase-only parallel-aligned nematic LC SLMs (Holoeye Photonics AG, Pluto), with 1920 x 1080 pixels, 8 μm pixel pitch and 60 Hz input image frame rate. The LC director of each SLM is arranged to be parallel to the direction of polarization of the incident light. The SLMs are used to display computer-generated phase pattern and modulate the two orthogonally polarized fields, respectively. Then, the modulated light beams are reflected by the SLMs and combined again with the help of NPBS to generate a SEB.

Fig. 1 An optical arrangement for producing SEB. L1, L2, lenses; HWP, half-wave-plate; NPBS, non-polarizing beam splitter; P1, P2, polarizers; SLM1, SLM2, spatial light modulators; PC1, PC2, computers.

Download Full Size | PDF

We utilize the MATLAB function rand to generate random grey-scale patterns, with uniform distribution. The random change of grey-scale color within the pattern creates a random change of phase. Dynamic graphs are created from a series of random grey-scale images, and then displayed on the PSLMs to impose a random phase fluctuation on the incident beam. Figure 2 shows one of the frames of the dynamic patterns generated with different modulation magnitude $γ$ of 0, 0.2, 0.5 and 1, respectively. One can find that the phase displayed on the SLMs has a larger range of values with larger modulation magnitude.

Fig. 2 One frame of dynamic patterns with modulation magnitude $γ$ of (a) 0, (b) 0.2, (c) 0.5, (d) 1. (see Visualization 1).

Download Full Size | PDF

As described above, the coherence degree of the x-polarized and y-polarized components of the SEB can be controlled by the modulation magnitudes. Dynamic graphs with different modulation magnitude were generated and displayed on the PSLMs. The modulation magnitudes were 0, 0.2, 0.4, 0.5, 0.6, 0.8, 0.9 and 1, respectively. By blocking one of them, we measure the coherence degree of the x and y components with Young’s double-pinhole interferometer, respectively. Each pinhole is a square hole with 0.1 mm side length, and the separation distance between the two pinholes is 1.2 mm. A digital camera (Ophir, SP620U) is used to record the interference pattern. For each modulation magnitude of dynamic graph, 90 interference patterns were recorded in order to calculate the coherence degree. The results in Fig. 3(a) are the average values.

Fig. 3 (a) Filled circles and diamonds represent the measured $μ_{x x}$ and $μ_{y y}$ , respectively; Short dash line and solid line represent the fitting curve of $μ_{x x}$ and $μ_{y y}$ ; Dotted line represents the theoretical simulation of $μ_{j j}$ . (b) Measured M² of the x-polarized light versus different modulation magnitude.

Download Full Size | PDF

From Fig. 3(a) it is clearly seen that, both the coherence degree of the x component ( $μ_{x x}$ ) and that of the y component ( $μ_{y y}$ ) decrease with the increase of the modulation magnitude, that is, the range of the phase modulation. The difference between the measured results may be caused by the minor individual differences between the two SLMs. We fit the experimental result with a function of $μ_{j j} = a * {sinc}^{2} (γ_{j}) + b$ . The fitting coefficients we used are a = 0.6552, b = 0.3368 for $μ_{x x}$ and a = 0.6821, b = 0.311 for $μ_{y y}$ respectively. As shown in Eq. (10), the coherence degree of the SEB’s polarization components is given by $μ_{j j} = {sinc}^{2} (γ_{j})$ . The dotted line shown in Fig. 3(a) represents the ideal theoretical result. There may be some possible reasons that cause the difference between the measured results and the theoretical result, i.e. the arising of the parameters a and b in the fitting result. Firstly, the response speed of the PSLMs is too slow to achieve complete incoherent light in our experiments. Secondly, the size of the pinhole (0.1 mm) is much larger than that of the PSLM’s pixel (8 μm). It means that the light enters each pinhole is composed of the light field of many points. Last but not the least, in theory the light source is assumed to be ergodic. However, the random phase we generated in experiments by MATLAB is discrete but not continuous. Nevertheless, the experimental result confirms that we are able to control the coherence degree $μ_{x x}$ and $μ_{y y}$ , independently on each other.

The beam propagation factor (also known as the M²-factor) proposed by Siegman [27] is a particularly useful property of an optical laser beam, and it plays an important role in the characterization of beam behavior on propagation. The definition of M²-factor in terms of second-order moments of the Wigner distribution function has been adopted widely for characterizing laser beams [28–30]. The second-order moments are given in terms of CSD function. Therefore, we measure the M²-factor of the x-polarized light with a commercial M² measuring instrument (Ophir, M2-200S). One observes from Fig. 3(b) that the value of the M²-factor of the x-polarized light increases as the modulation magnitude increases, while the coherence degree decreases with the increase of the modulation magnitude, as shown in Fig. 3(a).

Then, we blocked the y-polarized component and displayed dynamic graph generated with modulation magnitude of 0.5 on SLM2. We observed the interference pattern of the x-polarized component through double-pinholes with different separation distances. The distances between the two pinholes are 0.5, 0.7, 0.9, 1, 1.2, 1.4 and 1.8 mm, respectively. For each distance between the two pinholes, 90 interference patterns were recorded in order to calculate the coherence degree. The results in Fig. 4 are the average values. Figure 4 shows that the coherence degree does not depend on the distance between the pinholes, exhibiting good correspondence with the prediction in Section 2.

Fig. 4 Coherence degree versus the distance of the two pinholes.

Download Full Size | PDF

The recombination of the two orthogonally polarized partially coherent beams with the help of the NPBS will generate a SEB. In order to measure the coherence degree between x - and y - polarized components of the SEB, namely $| μ_{x y} |$ , we employ a modified Mach-Zehnder interferometer as shown in Fig. 5. The polarizing beam-splitter (PBS) separates the x and y components by reflecting the y component, while allowing the x component to pass through. We convert the y component to be also x polarized with a half-wave-plate (HWP). We place two knife-edges in each arm of the interferometer. As a result, the knife-edge (A1) blocks half of the beam in the lower arm, and the other one (A2) blocks the other half in the upper arm. The two arms are combined by a NPBS and incident on a Young’s double-pinhole. We impose various random phases with different modulation magnitude $γ_{x}$ and $γ_{y}$ on the SLMs to generate various SEBs and measure the coherence degree. For each group of $γ_{x}$ and $γ_{y}$ , 90 interference patterns were recorded in order to calculate the coherence degree. The results in Fig. 6 are the average values. Figure 6 shows the experimental results of coherence degree as a function of modulation magnitude $γ_{x}$ while $γ_{y}$ remains unchanged. One can find that, the coherence degree $μ_{x y}$ decreases with the increasing of $γ_{x}$ for fixed $γ_{y}$ . The two dash straight lines shown in Fig. 6 indicate the measured result of $μ_{x x}$ when $γ_{x} = 0.4, 0.6$ respectively. It should be noted that the measured value of $μ_{x y}$ for $γ_{x} = γ_{y} = 0.4 or 0.6$ is slightly smaller than that of $μ_{x x}$ . This discrepancy may be caused by the minor individual differences between the two SLMs, and also by the difference between the optical path from the NPBS to SLM1 and that to SLM2.

Fig. 5 Experimental setup of modified Mach-Zehnder interferometer. PBS, polarizing beam-splitter. A1, A2, knife-edges. M1, M2, mirrors.

Download Full Size | PDF

Fig. 6 Coherence degree as a function of the modulation magnitude $γ_{x}$ when $γ_{y}$ is equal to (a) 0.4 and (b) 0.6, respectively.

Download Full Size | PDF

We also measure the beam propagation factor M² of the SEBs. The four subfigures of Fig. 7 show the dependence of the measured M²-factor on the modulation magnitude $γ_{x}$ , when $γ_{y} = 0, 0.4, 0.6, 1$ , respectively. It should be noted that the scale in Fig. 7(a) is different from that of the others. As expected, when the modulation magnitude $γ_{y}$ keeps constant, the M² of the SEB increases with an increase in $γ_{x}$ . These results lead to the conclusion that the coherence degree of the SEB can be controlled by making the appropriate choice of the modulation magnitudes $γ_{x}$ and $γ_{y}$ .

Fig. 7 M² versus modulation magnitude $γ_{x}$ when $γ_{y}$ is equal to (a) 0, (b) 0.4, (c) 0.6, and (d) 1, respectively.

Download Full Size | PDF

4. Conclusion

We report the generation of a stochastic electromagnetic beam (SEB) with complete controllable coherence. We provide an effective method for quantitative and real-time control of the beam’s coherence degree. The coherence properties of the beam are controlled by varying the modulation magnitudes of the random phase, which we applied onto two crossed phase only SLMs. We measure the beam’s coherence properties as well as the beam propagation factor. The experimental results confirm that the coherence degree can be independently controlled along two mutually perpendicular directions. Moreover, the coherence degree of the SEB we have generated is independent on both the positions and the distance between any two points. In particular, the spatial correlation structure of the SEB can be also controlled, if the modulation magnitudes are dependent on positions.

Funding

National Natural Science Foundation of China (NSFC) (61575070, 61605049, 11304104); Fujian Province Science Funds for Distinguished Young Scholar (2015J06015); Natural Science Foundation of Fujian Province of China (2014J05007).

Acknowledgments

The authors would like to thank Professor Sabino Chavez-Cerda for helpful discussion and valuable assistance in language editing.

References and links

1. Y. Wang, P. Meng, D. Wang, L. Rong, and S. Panezai, “Speckle noise suppression in digital holography by angular diversity with phase-only spatial light modulator,” Opt. Express 21(17), 19568–19578 (2013). [CrossRef] [PubMed]

2. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19(8), 1592–1598 (2002). [CrossRef] [PubMed]

3. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002). [CrossRef] [PubMed]

4. J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8(2), 282–289 (1991). [CrossRef]

5. J. Rosen and M. Takeda, “Longitudinal spatial coherence applied for surface profilometry,” Appl. Opt. 39(23), 4107–4111 (2000). [CrossRef] [PubMed]

6. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

7. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]

8. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef] [PubMed]

9. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef] [PubMed]

10. H. Lajunen and T. Saastamoinen, “Propagation characteristics of partially coherent beams with spatially varying correlations,” Opt. Lett. 36(20), 4104–4106 (2011). [CrossRef] [PubMed]

11. S. Cui, Z. Chen, L. Zhang, and J. Pu, “Experimental generation of nonuniformly correlated partially coherent light beams,” Opt. Lett. 38(22), 4821–4824 (2013). [CrossRef] [PubMed]

12. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

13. F. Gori, V. Ramírez-Sánchez, M. Santarsiero, and T. Shirai, “On genuine cross-spectral density matrices,” J. Opt. A, Pure Appl. Opt. 11(8), 85706 (2009). [CrossRef]

14. Z. Mei, O. Korotkova, and E. Shchepakina, “Electromagnetic multi-Gaussian Schell-model beams,” J. Opt. 15(2), 25705 (2013). [CrossRef]

15. Z. Tong and O. Korotkova, “Electromagnetic nonuniformly correlated beams,” J. Opt. Soc. Am. A 29(10), 2154–2158 (2012). [CrossRef] [PubMed]

16. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef] [PubMed]

17. T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005). [CrossRef]

18. A. S. Ostrovsky, G. Martínez-Niconoff, V. Arrizón, P. Martínez-Vara, M. A. Olvera-Santamaría, and C. Rickenstorff-Parrao, “Modulation of coherence and polarization using liquid crystal spatial light modulators,” Opt. Express 17(7), 5257–5264 (2009). [CrossRef] [PubMed]

19. E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003). [CrossRef]

20. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

21. L. Burger, I. Litvin, S. Ngcobo, and A. Forbes, “Implementation of a spatial light modulator for intracavity beam shaping,” J. Opt. 17(1), 015604 (2015). [CrossRef]

22. J. Tervo, T. Setälä, and A. T. Friberg, “Phase correlations and optical coherence,” Opt. Lett. 37(2), 151–153 (2012). [CrossRef] [PubMed]

23. T. Shirai and E. Wolf, “Coherence and polarization of electromagnetic beams modulated by random phase screens and their changes on propagation in free space,” J. Opt. Soc. Am. A 21(10), 1907–1916 (2004). [CrossRef] [PubMed]

24. B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5(3), 191–192 (1963). [CrossRef]

25. J. Tervo, T. Setala, and A. Friberg, “Degree of coherence for electromagnetic fields,” Opt. Express 11(10), 1137–1143 (2003). [CrossRef] [PubMed]

26. P. Réfrégier and F. Goudail, “Invariant degrees of coherence of partially polarized light,” Opt. Express 13(16), 6051–6060 (2005). [CrossRef] [PubMed]

27. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE 1224, 2–14 (1990).

28. B. Zhang, X. Chu, and Q. Li, “Generalized beam-propagation factor of partially coherent beams propagating through hard-edged apertures,” J. Opt. Soc. Am. A 19(7), 1370–1375 (2002). [CrossRef] [PubMed]

29. X. Ji, T. Zhang, and X. Jia, “Beam propagation factor of partially coherent Hermite-Gaussian array beams,” J. Opt. A, Pure Appl. Opt. 11(10), 105705 (2009). [CrossRef]

30. G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. A, Pure Appl. Opt. 12(1), 015701 (2009).

Generation of stochastic electromagnetic beams with complete controllable coherence

Abstract

1. Introduction

2. Theoretical analysis

3. Experimental setup and results

4. Conclusion

Funding

Acknowledgments

References and links

Supplementary Material (1)

Cited By

Figures (7)

Equations (16)

Optics Express